EXTREMAL ORIENTED GRAPHS AND
ERDOS-HAJNAL˝ CONJECTURE
by
Payam Valadkhan
B.Sc., Sharif University of Technology, 2007
a Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the School of Computing Science
c Payam Valadkhan 2009 SIMON FRASER UNIVERSITY Summer 2009
All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL
Name: Payam Valaclkhan
Degree: N{aster o[ Science
Title of Thesis: Extrernal OrierrteclGraphs and Erd6s-Hajnal Conjecture
Examining Committee: Dr. RarrreshKrisirnamurti Cirair
Dr. Gd,bor Tardos, Professor',Compnting Science Sirnori Fra,serUniversity Seriior Supervisor
Dr. Arrrlrei Bulat,ov. Associate Professor, Ooniputing Scierrce SirrronF'raser L rriversitv Superrvisor
Dr. LaclislarvStacho. Associate Professor. N,{at}renriit- ics Simorr Fraser University SFU Exanriner
Date Approved: /ll,tnp vl tll- 4'1-lq| ./ ilu ii^rl- )r,C,') ,/
!,t
Declaration of Partial Copyright Licence
The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users.
The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the “Institutional Repository” link of the SFU Library website
The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies.
It is understood that copying or publication of this work for financial gain shall not be allowed without the author’s written permission.
Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence.
While licensing SFU to permit the above uses, the author retains copyright in the thesis, project or extended essays, including the right to change the work for subsequent purposes, including editing and publishing the work in whole or in part, and licensing other parties, as the author may desire.
The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive.
Simon Fraser University Library Burnaby, BC, Canada
Last revision: Spring 09 Abstract
For a (finite or infinite) family L of oriented graphs, a new parameter called the compress- ibility number of L and denoted by z(L) is defined. The main motivation is the application of this parameter in a special case of Tur´an-type extremal problems for digraphs, in which it plays the role of chromatic number in the classic extremal problems. We estimate this parameter for some special group of oriented graphs. Determining this parameter, in the most explicit possible form, for oriented graphs with bounded oriented coloring number (planar graph in particular) leads us to the famous Erd˝os-Hajnal conjecture.
iii Contents
Approval ii
Abstract iii
Contents iv
1 Introduction 1 1.1 Background and old results ...... 1 1.2 New results and the thesis structure ...... 5 1.3 Notations ...... 6
2 The nature of compressibility number 7 2.1 The existence ...... 7 2.2 Oriented Erd˝os-Stone-Simonovits ...... 7
3 Compressibility number for some oriented graphs 13 3.1 Sparse graphs ...... 13 3.2 Dense graphs ...... 16
4 Relationship with Erd˝os-Hajnal conjecture 19
5 Some oriented Erd˝os-Hajnal problems 25 5.1 Partitioning restrictions ...... 25 5.2 A constructive method ...... 27
Bibliography 30
iv Chapter 1
Introduction
1.1 Background and old results
By digraph we mean a directed graph which contains no loop and no multi-arcs: two or more arcs with the same direction between a pair of vertices. But it can have two opposite arcs between a pair of vertices, which are called symmetric arcs. For any digraph H we denote by V (H) and E(H), the set of vertices and arcs of H, respectively. The same definitions apply when replacing the notion digraph and arc with simple graph and edge, respectively. From now on by graph we mean simple graph. For x, y ∈ V (H), we write xy−→ to indicate that there is an arc in E(H) oriented from x to y. The same notation also represents the arc itself. A digraph H is called a sub-digraph of another digraph H0, and denoted by H ⊆ H0, if V (H) ⊆ V (H0) and E(H) ⊆ E(H0). For S ⊆ V (H0), we say the 0 0 digraph H is called the induced sub-digraph of S in H , and denote by H |S, if V (H) = S and E(H) consists of all members of E(H0) with their endpoints in S.
A homomorphic mapping f from a digraph H to another digraph H0 is a mapping f : −−−−−−→ V (H) → V (H0) such that f(x)f(y) whenever xy−→. In this case we write H → H0 and say H is homomorphic to H0. Should the mapping f be a bijection with its inverse function being also a graph homomorphic mapping, then f is called a graph isomorphic mapping and we say H is isomorphic to H0. We say that H0 contains H if H is isomorphic to a sub-digraph of H0, otherwise we say H0 avoids H or, equivalently, H0 is H-free.
1 CHAPTER 1. INTRODUCTION 2
In their joint paper, [5], Brown and Harary initiated the study of extremal digraphs by considering the digraph analogues of the Paul Tur´an’sclassic extremal theorem [19, 20]: What is the maximum number of arcs in a digraph avoiding a certain forbidden digraph? And what are such extremal digraphs? Their study gives precise answers for the forbidden digraphs with four or less vertices.
Later on Brown, Erd˝osand Simonovits, in a series of papers [2, 3, 4], studied this problem and its extension to any family (finite or infinite) of forbidden digraphs. Given integer n ≥ 1 and a square matrix A with zero or one along the main diagonal and zero or two anywhere else, in [2], they introduce a method for constructing a digraph A(n) where |V (A(n))| = n. Then they show that for any family (finite or infinite) of forbidden digraphs, there is a matrix A such that for every n ≥ 1, A(n) contains non of the forbidden digraphs and |E(A(n))| is very close to its optimal. Thus, this study ensures the existence of such a family of approximately extremal digraphs and gives some insight into its structure. But it does not answer the digraph version of the Paul Tur´an’sextremal theorem, i.e., a clear estimation for the number of arcs of the extremal digraphs with n vertices. This is due to the unknown order of the matrix A which is used in the construction.
Attempts were made to find some estimations for this value. In [4] an algorithm is intro- duced to calculate it. In [16] some upper bounds were proved in terms of some parameters of the forbidden digraphs. These studies revealed the inherent complexity for finding any meaningful estimation of the number of extremal digraphs’s arcs. This is in contrast to the undirected version in which a result of Erd˝os,Stone and Simonovits [9] gives us an exact asymptotic value for the number of edges of the extremal graphs in terms of the chromatic numbers of the forbidden graphs.
The situation becomes more tractable if we limit ourselves to the special case of oriented graphs: a family of digraphs which contain no pair of symmetric arcs. Actually this is suggested by Brown and Harary themselves in the last section of their paper [5] as a possible direction for future works. But this case, which we call the extremal theory of oriented graphs, was not considered in the later works of Brown et al. [2, 3, 4] and other subsequent works [16]. CHAPTER 1. INTRODUCTION 3
The work in the present thesis is along this unstudied direction. We say a oriented graph H is an orientation of a graph G if it is obtained from G by assigning an orientation to each edge of G. If H is acyclic (contains no directed cycle), then we say H is an acyclic orientation of G. An orientation of a complete graph is called tournament. An acyclic tournament is called transitive tournament. From extremal theory’s point of view, the chromatic number of a graph G can be re-defined as the smallest number k such that G is homomorphic to the complete graph on k vertices. This definition can be naturally reformulated for oriented graphs by replacing the notion graph with the notion oriented graph, however the term ’complete graph on k vertices’ cannot be simply replaced with ’complete oriented graph (=tournament) on k vertices’ as we have many such tournaments. Thus the oriented analogues of this definition, after generalizing to a family of oriented graphs, will read:
Definition 1.1.1. For any finite or infinite family L of oriented graphs, z(L), the compress- ibility number of L, is the smallest number k such that for any tournament on k vertices, at least one member of L is homomorphic to it. When L has only one member H, we may use the notations z(L) and z(H) interchangeably.
Later in Theorem 2.1.1 we will see that z(L) is well-defined, should L contain at least one acyclic oriented graph. We will show (Theorem 2.2.8) that this chromatic-type parameter replaces the role of chromatic number in the extremal oriented graph problem.
Definition 1.1.2. For any oriented graph H, we define p(H) to be the length (number of edges) of the longest directed path in H.
One may find the definition of z(H) similar to that of one-color oriented Ramsey number r(H) (for example refer to [12]). r(H) is defined to be the smallest integer k such that H is isomorphic to any tournament on k vertices. It must be noted that in calculating the Ramsey number r(H), we look for an isomorphic copy of H in any arbitrary tournament while in the case of z(H) we look for a homomorphic copy of H. This makes a huge difference as we will see. Thus, the known results for oriented Ramsey theory (see [12] for a list of references) cannot contribute significantly to the compressibility number as the concept of isomorphism is different from homomorphism except in some rare cases. One such case is when H is a transitive tournament. In this case a well known result of Erd˝os-Moser[8] (re- stated in Theorem 3.2.1) gives some estimation for the values of z(H). Presently, this is the CHAPTER 1. INTRODUCTION 4 only remarkable contribution of oriented Ramsey theory in determining a compressibility number.
Motivated by these relations we embark on determining z(L) for any family of oriented graphs. Here we see that, in contrast to the oriented Ramsey problem which is hard even for orientations of paths and cycles (see [12]), compressibility number is more tractable. By employing various graph theoretic concepts (like chromatic number, planarity, etc) we well estimate z(H) for some large and important groups of oriented graphs H, including orien- tations of trees (Theorem 3.1.1), cycles (Theorem 3.1.3), unions of internally disjoint paths with the same endpoints (Theorem 3.1.6). In these cases, the value of z(H) is expressed in terms of p(H). Additionally some estimations are given for acyclic orientations of complete graphs (Theorem 3.2.1) and complete bipartite graphs (Theorems 3.2.4,3.2.5).
The oriented coloring number of an arbitrary oriented graph H, denoted by χo(H), is the smallest number k ≥ 0 such that H is homomorphic to an oriented graph on k vertices. Inspired by these results we target a more general family of oriented graphs which encom- passes the previous ones: When H is an orientation of a planar graph. Based on the previous results we conjecture that z(H) is upper bounded by p(H)d for some constant d (Conjecture 3.1.9). We show a direct connection (Theorem 4.0.17) between this conjecture and the so- called oriented Erd˝os-Hajnal conjecture (re-stated in Conjecture 4.0.10). Introduced in [1], this conjecture is about the existence of a transitive sub-tournament with polynomial order in any tournament which avoids a forbidden oriented graphs. We show (Corollary 4.0.18) that the oriented Erd˝os-Hajnal conjecture implies our planar graph conjecture and even a more general conjecture claiming the same upper bound when H has bounded oriented coloring number (of which planar graph orientations are a special case, see [14]).
This connection enables us to come up with good estimations of z(H) in terms of p(H), should we validate the oriented Erd˝os-Hajnal conjecture, at least for some certain forbidden oriented graphs (Theorem 5.0.22). This makes further motivation to study this conjecture . A result in [1] states that this conjecture is equivalent to the classic Erd˝os-Hajnal conjecture (re-stated in Conjecture 4.0.9) for graphs [7]. Moving along this direction we try to come up with the oriented analogous of some Erd˝os-Hajnal related problems stated in [7, 1] (Theorems 4.0.12,4.0.20,5.2.3). The result of this analysis is the validation of the oriented Erd˝os-Hajnal conjecture for some special classes of forbidden oriented graphs (Corollaries CHAPTER 1. INTRODUCTION 5
5.1.5,5.1.6) along with a pseudo-polynomial upper bound in terms of p(H) for z(H), when H has bounded oriented number (Corollary 4.0.21).
1.2 New results and the thesis structure
A simple argument (Theorem 2.1.1) in the first part of Chapter 2 ensures the existence of z(L) with the condition that L contains at least one acyclic oriented graph. The famous Erd˝os-Stone-Simonovits theorem [9] (re-stated in Theorem 2.2.2) states that the maximum number of edges of a graph on n vertices avoiding a family of forbidden graphs is asymptot- r−2 n ically r−1 2 with r being the minimum chromatic number among the forbidden graphs. In the second part of Chapter 2 we show the oriented version of this theorem (Theorem 2.2.8) with the chromatic number r substituted with the compressibility number parameter: the maximum number of arcs of any oriented graph on n vertices which avoids all member of is asymptotically (Definition 2.2.3): z(L)−2 n. Furthermore the optimal matrix for L z(L)−1 2 L (Definition 2.2.4) has order z(L) − 1 and the oriented graphs generated by this matrix are certain orientations of the Tur´angraphs with z(L) − 1 parts (Theorem 2.2.8).
In Chapter 3 we well-estimate the value of z(H), in terms of p(H) in most cases, when H is any orientation of the following classes of graphs: trees (Theorem 3.1.1), cycles (Theorem 3.1.3), unions of internally disjoint paths with shared endpoints (Theorem 3.1.6), complete (Theorem 3.2.1) and complete bipartite graphs (Theorem 3.2.4). Based on these results we conjecture that in general z(H) is upper bounded by p(H)d for some constant d when H is any acyclic orientation of a planar graph (Conjecture 3.1.9).
Chapter 4 is dedicated to the latter conjecture. We show a close connection between the compressibility number and the order of the biggest transitive sub-tournament in a tourna- ment avoiding some forbidden tournament (Theorem 4.0.17). This way we link (Corollary 4.0.18) our conjecture for the planar graph orientations and a more general class of graphs (i.e. the class of oriented graphs with bounded oriented coloring numbers) with the oriented version of Erd˝os-Hajnal conjecture formulated in [1] (re-stated in Conjecture 4.0.10).
In Chapter 5 we study some Erd˝os-Hajnal related problems motivated by the results of the previous chapter. With a very straightforward argument we convert some results from [7] (re-stated in Theorem 4.0.19) and [1] (re-stated in Theorem 5.2.2) to their oriented versions CHAPTER 1. INTRODUCTION 6
(Theorems 4.0.20,5.2.3). The conversion of the first theorem (Theorem 4.0.20) ensures a transitive sub-tournament with pseudo-polynomial order for restricted tournaments. This leads to the result that z(H) is upper bounded by p(H)d log p(H) (for some constant d) for any oriented graph H with bounded oriented coloring number (Corollary 4.0.21). The conversion of the second theorem (Theorem 5.2.3) gives us a very powerful blow-up construction method to validate the oriented Erd˝os-Hajnal conjecture for more oriented graphs composed of the known ones. We prove this conjecture for any oriented graph of order at most four as well as any directed cycle as the forbidden oriented graph. Armed with these results we can validate the conjecture for even more oriented graphs, i.e., all tournaments with the length of its biggest directed cycle at most four (Corollary 5.2.4).
1.3 Notations
For an oriented graph H, we denote by |V (H)| the order of H. Let v ∈ V (H), we denote by N +(v) and N −(v) the set {u ∈ V (H)|vu−→} and {u ∈ V (H)|uv−→}, respectively. |N +(v)| and |N −(v)| are called out-degree and in-degree of v in H, respectively. The sum of in-degree and out-degree of v is called the degree of v. The single arc oriented graph is an oriented graph with two vertices and an arc between them. A directed walk W = (v0, v1, ··· , vl) of order l in H is defined as a sequence of vertices of H (not necessarily distinct) such that for −−−→ every 0 ≤ i < l, vivi+1. We call v0 and vl the starting and ending vertex of W , respectively. At some occasions during this work, we may treat W as a set of its vertices (for instance when we define a mapping on W or talk about its membership), in this occasions consider it as the set {v0, v1, ··· , vl}. We say W is a directed path if all of its vertices are distinct. A directed path with an additional arc going from its ending vertex to its starting vertex is called a directed cycle. Any orientation of a path is called oriented path. Chapter 2
The nature of compressibility number
2.1 The existence
In this section we show that when z(L) is well-defined:
Theorem 2.1.1. a) For any finite or infinite family L of oriented graphs, z(L) has a finite value if and only if L has at least one acyclic oriented graph. b) Given an acyclic oriented graph H, z(H) ≥ p(H) + 1
Proof. a) If L contains no acyclic oriented graph, then every transitive tournament avoids all its members, so z(L) if not defined. Now suppose H ∈ L is an acyclic oriented graph. Then according to Erd˝os-Mosertheorem [8], H is homomorphic to any tournament of order |V (H)|−1 |V (H)|−1 2 . This means z(L) is upper bounded by 2 . b) Obviously H cannot be homomorphic to the transitive tournament of order p(H), as the latter has no oriented walk of order p(H) + 1, so the inequality follows.
2.2 Oriented Erd˝os-Stone-Simonovits
In this section we show the role of compressibility number in extremal oriented graph theory.
Definition 2.2.1. For any family L of graphs and n ≥ 1, we define ex(n, L) to be the biggest integer k such that there is a graph G with n vertices and k edges which has no member of
7 CHAPTER 2. THE NATURE OF COMPRESSIBILITY NUMBER 8
L as sub-graph. All such graphs G which yield this maximum are called extremal graphs. We define ex(n, L) similarly with the notions graph and edge replaced with digraph and arc, respectively. We define exo(n, L) similar to ex(n, L) with the extra condition of digraph G being an oriented graph.
Erd˝os-Stone-Simonovits theorem ([9]) gives the following asymptotic estimation for ex(n, L):
Theorem 2.2.2. For any family L of graphs, let r = minF ∈L{χ(F )}. Suppose r ≥ 2. Then r − 2 n ex(n, ) = + o(1) (2.1) L r − 1 2 where o(1) → 0 as n → ∞
Finding a similar estimation for ex(n, L) is more challenging (at least up until now no good estimation is found):
Definition 2.2.3. ([2]) The sequence S = (H1,H2, ···) of digraphs with |V (Hn)| = n for every n ≥ 1, is called a sequence of asymptotic extremal digraphs for L, if no member of S contains any member of L and: |E(H )| lim n = 1. n→∞ ex(n, L)
Definition 2.2.4. ([2]) Given n ≥ 1 and a matrix A = (ai,j)i,j≤r with 0 or 1 along the main diagonal and 0 or 2 outside of the diagonal. The optimal matrix graph A(n) is a directed graph on n vertices with maximum number of arcs such that V (A(n)) can be partitioned into −→ r classes C1,C2, ··· ,Cr satisfying two conditions: First, for all 1 ≤ i 6= j ≤ r: xy for every x ∈ Ci and y ∈ Cj if and only if ai,j = 2, second, for all 1 ≤ i ≤ r: The induced sub-digraph of Ci is a transitive tournament if ai,i = 1 and an empty graph (no arcs) otherwise. If there are several choices for A(n), pick one of them arbitrary.
Now the main result of [2] concerning the value ex(n, L) is as follows:
Theorem 2.2.5. ([2]) For any finite or infinite family L of digraphs there exists a matrix A = (ai,j)i,j≤r such that {A(n)}n≥1 is a sequence of asymptotic extremal digraphs for L.
This result and the techniques used in its proof are not used anywhere in this thesis. So I do not include its long full proof here and instead, I give a general outline of the proof. Please refer to the original paper [2] for the complete proof. CHAPTER 2. THE NATURE OF COMPRESSIBILITY NUMBER 9
Definition 2.2.6. For an n × n matrix A we define g(A), the density of the matrix A, as −1 max{uAu } where the maximum is taken over all vectors u = (u1, u2, ··· , un) with ui ≥ 0 Pn and i=1 ui = 1.
Proof. (Theorem 2.2.5, A short summary from [2]) Given a matrix A and a family of digraphs n n {G } , we shall say that A is weakly contained by the family {G } if the maximum m = mn n for which A(m) ⊆ G is unbounded as n → ∞. Suppose L = {L1,L2, · · ·}. First, a n sequence {Z }n=1,2,··· of digraphs with following extremal property is defined: for each n, n Z avoids the members of L and each of its vertices has degree ≥ d where d is the maximum integer for which such a sequence of digraphs exists. The aim is to construct a matrix B n which is weakly contained in {Z }n=1,2,··· and has maximum density. They show that this condition guarantees {B(n)}n=1,2,··· to be a sequence of asymptotic extremal digraphs. To find such a matrix B, they use the following recursive construction: Suppose the matrix A n is weakly contained by the family {Z } and g(A) < lim supn→∞ dn/n (dn is the minimum degree in Zn). Then they construct (in a lemma) a matrix A0 which is weakly contained in n 0 {Z }n=1,2,··· and g(A ) > g(A). Now they start from the matrix Dr = (2 − δij)i,j≤r where
δij is the Kronecker symbol: 1 if i = j and 0 otherwise and r is the largest integer such that n Dr is weakly contained in {Z }n=1,2,···. They show that such an r exists. At each step this construction is applied to the matrix obtained in the last step, until we arrive at a matrix
B where g(B) = lim supn→∞ dn/n. They show that this matrix is obtainable in finite steps. n Note that B(n) avoids all the members of L since B is weakly contained in {Z }n=1,2,··· and n Z avoids all the members of L. Next they show that the condition g(B) = lim supn→∞ dn/n indeed grantees its being a sequence of asymptotic extremal digraphs.
In our attempt to estimate the value ex(n, L), we need the oriented graph version of Definition 2.2.3:
Definition 2.2.7. The sequence S = (H1,H2, ···) of oriented graphs with |V (Hn)| = n for every n ≥ 1, is called a sequence of asymptotic extremal oriented graphs for L if no member of S contains any member of L and: |E(H )| lim n = 1. n→∞ exo(n, L) We formulate the oriented graph version of Theorem 2.2.5 which contains more infor- mation about the structure of matrix A owing to the simpler structure of oriented graphs CHAPTER 2. THE NATURE OF COMPRESSIBILITY NUMBER 10 compared to digraphs. This extra clarification helps us to come up with a clear asymptotic estimation for exo(n, L) which is similar to that of Theorem 2.2.2:
Theorem 2.2.8. a) For any finite or infinite family L of oriented graphs, with at least one acyclic member with more than one arc and z(L) ≥ 2, there exists a matrix A = (a ) such that {A(n)} is a sequence of asymptotic extremal oriented graphs for i,j i,j≤z(L)−1 n≥1 L. b) Furthermore the matrix A has two properties: first, all the entries along the main diagonal are zero and second, ai,j + aj,i = 2 for every 1 ≤ i 6= j ≤ z(L) − 1. In other words, A(n) (n ≥ 1) is an orientation of the Tur´angraph T . Thus: n,z(L)−1 z(L) − 2 n exo(n, L) = + o(1) (2.2) z(L) − 1 2 where o(1) → 0 as n → ∞
The comparison of (2.2) in Theorem 2.2.8 to (2.1) in Theorem 2.2.2 implies that com- pressibility number is the exact substitute for chromatic number when it comes to extremal problems for graphs and oriented graphs, respectively. The part (a) of Theorem 2.2.8 is ob- tained by applying Theorem 2.2.5 to the family L ∪ {A} of digraphs where A is the digraph consisting of two vertices and a pair of symmetric arcs between them. So our proof mainly targets part (b). We need a Ramsey-type argument before proving this theorem:
Lemma 2.2.9. There exists an unbounded non-decreasing function f : N → N such that, for every n ≥ 1 and an arbitrary orientation H of a complete bipartite graph, with parts A and B having n vertices each, there exist subsets A1 ⊆ A and B1 ⊆ B such that |A1| = |B1| = f(n) and all the arcs between A1 and B1 have the same orientation.
Proof. Suppose n is large enough such that m = blog n − log log nc < n (define f(n) = 1 for small ns). Choose m arbitrary vertices v1, v2, ··· , vm in part A. Make the sets Cm ⊆
Cm−1 ⊆ · · · ⊆ C1 ⊆ C0 recursively as follows: Set C0 = B, for any m ≥ i ≥ 1, Ci + − is either N (vi) ∩ Ci−1 or N (vi) ∩ Ci−1 depending on which one has bigger size. Then m obviously |Cm| ≥ |B|/2 ≥ log n. For each vi, all the arcs between this vertex and Cm have the same orientation. Then according to the Pigeon hole principle we can find a subset
D ⊆ {v1, v2, ··· , vm} with |D| ≥ m/2 such that, all the arcs between D and Cm have the same orientation. Since these sets both have the size larger than m/2 we can safely set f(n) = bm/2c. CHAPTER 2. THE NATURE OF COMPRESSIBILITY NUMBER 11
Proof. (Theorem 2.2.8) Let T be a tournament on z( ) − 1 vertices avoiding all the z(L)−1 L members of . Let V (T ) = {v , v , ··· , v }. Let A = (a ) be a matrix L z(L)−1 1 2 z(L)−1 i,j i,j≤z(L)−1 −−→ with zero along the main diagonal and ai,j = 2 if vivj and zero otherwise. We claim that this is the desirable matrix. Given an arbitrary n ≥ 1. Let C ,C , ··· ,C be 1 2 z(L)−1 the vertex classes of V (A(n)) as defined in Definition 2.2.4. Suppose for some H ∈ L, H → A(n). Let f : V (H) → V (A(n)) be the corresponding homomorphic mapping. Define the homomorphic mapping g from H to T as follows: for every h ∈ V (H) let i be the z(L)−1 index such that, f(h) ∈ Ci, then define g(h) = vi. This mapping contradicts the definition of T . Thus A(n) avoids all members of . Let B be a (z( ) − 1) × (z( ) − 1) matrix z(L)−1 L L L such that {B(n)}n≥1 is a sequence of asymptotic extremal oriented graphs for L. Suppose {B(n)}n≥1 is not a sequence of asymptotic extremal oriented graphs for L. Then we will have: |E(B(n))| lim sup = b > 1 (2.3) n→∞ |E(A(n))|
This implies that there is an infinite set N1 ⊆ N such that: |E(B(n))| lim = b > 1 (2.4) n∈N1→∞ |E(A(n))|
Note that according to the Tur´an’stheorem [19, 20] A(n), in order to have the maximum number of arcs, should be an orientation of the Tur´angraph T . Pick a large enough n,z(L)−1 n ∈ N1 (later in the proof we will see how large). According to Erd˝os-Stonetheorem ([10]), (2.4) implies that there exists an unbounded monotone function t : N → N such that, B(n) contains a certain orientation of the complete z(L)-partite graph with t(n) vertices in each part. Call this oriented graph W and its parts W ,W , ··· ,W . Run the following 1 2 z(L) algorithm: Consider an arbitrary order for all the pairs (i, j) with 1 ≤ i < j ≤ z(L). Then according to this order for each pair (i, j), in its turn, replace the sets Wi and Wj with the sets A1 and B1 respectively where the last two sets are obtained by applying Lemma
2.2.9 to A = Wi and B = Wj. Let f be the function defined in this lemma. After this process, the original oriented graph W will have changed to a smaller oriented graph with all its arcs between every two parts W and W having the same orientation. Let T i j z(L) be a tournament on z( ) vertices w , w , ··· , w such that w−−−→w (i 6= j) iff the common L 1 2 z(L) i j orientation of the arcs between Wi and Wj in W , is from the latter to the former set. Then there exists H ∈ which possesses a homomorphic mapping f : V (H) → V (T ). L z(L) We can easily define a one-to-one homomorphic mapping g : V (H) → V (W ) as follows: CHAPTER 2. THE NATURE OF COMPRESSIBILITY NUMBER 12
for any h ∈ V (H) suppose f(h) = wi. Then assign an arbitrary vertex in Wi, which has not been assigned yet, to g(h). Such assignment is possible if n is large enough such that z( )−1 |Wi| = f L (t(n)) ≥ |V (H)| (recall f is the function defined in Lemma 2.2.9). Since both f and t are unbounded monotone functions, we can pick up a large enough n to fulfill the latter inequality. The consequence is, the mapping g introduces a sub-oriented graph of W isomorphic to H, in other words B(n) will contain H, contradiction. Thus the assumption stated in (2.3) is wrong and thus we conclude that {A(n)}n≥1 is a sequence of asymptotic extremal oriented graphs.
Open Question 2.2.10. Adopting the notations defined in this section, in the Erd˝os-
Simonovits theorem [9], the notion z(L) is replaced with minG∈L{χ(G)}. We conjecture that it is not true in the oriented version, i.e. z(L) 6= minH∈L{z(H)} for some family L of acyclic oriented graphs. Chapter 3
Compressibility number for some oriented graphs
3.1 Sparse graphs
The second part of theorem 2.1.1 gives p(H) + 1 as a lower bound for z(H). The results below show that for some sparse graphs, z(H) is very close, and sometimes equal to this minimum:
Theorem 3.1.1. Suppose H is an orientation of a tree, then z(H) = p(H) + 1 .
To prove this theorem and some others in this thesis, we need the following useful lemma:
Lemma 3.1.2. Assume H is an oriented acyclic graph, then it is homomorphic to transitive tournament with order p(H) + 1.
Proof. For every v ∈ V (H) define l(v) to be the length of the longest directed path ending at v. Obviously 0 ≤ l(v) ≤ p(H). Additionally if uv−→ then l(v) > l(u), since due to H being acyclic, the longest directed path P ending at u of length l(u) can be further extended to a directed path of length l(u) + 1 by adding the arc uv−→ to P . Now given the transitive −→ tournament T of order p(H) + 1 with the vertex sequence < t0, t1, ··· , tp(H) > where titj when i < j, define the following homomorphic mapping f from H to T : For all v ∈ H, f(v) = tl(v).
Proof. (Theorem 3.1.1) Let T be a tournament of order p(H) + 1. If T is transitive, then H → T according to Lemma 3.1.2. Otherwise T contains a directed cycle C of order 3.
13 CHAPTER 3. COMPRESSIBILITY NUMBER FOR SOME ORIENTED GRAPHS 14
One can easily see H → C (assign one of the labels 0, 1, 2 recursively to the vertices of H as follows: assign 0 to the root. For a vertex v if its parent w is assigned the label i, then −→ assign i + 1(mod 3) to v if wv and i − 1(mod 3) otherwise. Let c0, c1, c2 be vertices of C in −−−→ clockwise order (cici+1, i = 0, 1, 2, index calculations are mod 3). Then map all the vertices of H with label i [i = 0, 1, 2] to ci).
Theorem 3.1.3. Suppose H is an acyclic orientation of a cycle, then z(H) ≤ max{p(H) + 1, 6} .
Definition 3.1.4. Let (s, t) ∈ V (T ) × V (T ) be a pair of vertices in a tournament T . For l ≥ 0 we say (s, t) accepts a directed walk of order l if and only if there exists an order l directed walk W in T with s and t as its starting and ending vertex, respectively. According to this, define the accepting set of pair (s, t) as {l|(s, t)accepts a directed walk of order l} and denote it by DT (s, t) (note that if s = t then 0 ∈ DT (s, t))
Proof. (Theorem 3.1.3) V (H) can be arranged in a sequence A = (h0, h1, ··· , hn−1) such −−−−→ −−−−→ that, for 0 ≤ i < n, either hihi+1 or hi+1hi (index calculations are mod n). Since H −−→ −−−−→ itself is acyclic, w.l.o.g. we can assume h0h1 and h0hn−1. To each vertex hi (0 ≤ i < n), −−−−→ assign the letter ’f’ if hihi+1 and ’b’ otherwise. In this case h0 and hn−1 are assigned ’f’ and ’b’ respectively. Considering this fact, it’s easy to see that there exists a number t > 0 and 2t non-empty sequences F1,B1,F2,B2, ··· ,Ft,Bt of vertices of H such that
A = F1 ⊕ B1 ⊕ · · · Ft ⊕ Bt (operation ⊕ stands for sequence concatenation) and all vertices in Fi and Bi (1 ≤ i ≤ t) are assigned ’f’ and ’b’, respectively. Let Tm be a tournament with m = max{p(H) + 1, 6} vertices. Suppose there exist (s, t) ∈ V (Tm) × V (Tm) such that {1, 2, ··· , m − 1} ⊆ DTm (s, t), then we H → Tm, since we can define a homomorphic mapping f : V (H) → V (Tm) as follows: for every v ∈ V (H) consider two cases:
1. v ∈ Fi for some 1 ≤ i ≤ t
Suppose Fi = (u1, ··· , umi ) and v = uj. The definition of p(H) yields mi ≤ m − 1,
thus mi ∈ DTm (s, t), i.e. there is an order mi directed walk from s to t in Tm. Let
W = (w1, w2, ··· , wmi+1) be such a walk. Define f(v) = wj.
2. v ∈ Bi for some 1 ≤ i ≤ t
Suppose Bi = (u1, ··· , umi ) and v = uj. With the same argument as the previous
case we conclude that there is a walk W = (w1, w2, ··· , wmi+1) from s to t. This time
define f(v) = wmi+2−j. CHAPTER 3. COMPRESSIBILITY NUMBER FOR SOME ORIENTED GRAPHS 15
One can easily check that f is indeed a homomorphic mapping. This argument brings down our task to proving that there is indeed a pair (s, t) ∈ V (Tm)×V (Tm) such that {1, 2, ··· , m−
1} ⊆ DTm (s, t). It is easy to see the following additive property: For (s, t), (t, u) ∈ V (Tm) ×
V (Tm), DTm (s, t) + DTm (t, u) ⊆ DTm (s, u) (for two sets X and Y define X + Y = {x + y|x ∈ X, y ∈ Y }). Now we take the last step. Suppose Tm contains a directed cycle of length 3 and let v be one of its vertices. Then clearly {3k|k ≥ 0} ⊆ DTm (v, v). Suppose |N + (v)| ≥ |N − (v)| (the opposite case can be handled similarly). Let F be the oriented Tm Tm graph induced by N + (v)(N − (v)), then |V (F )| ≥ 3 since m ≥ 6. Thus F contains Tm Tm −→ −→ three distinct vertices x, y, z such that xy and yz. Then clearly DTm (v, z)(DTm (x, v)) contains {1, 2, 3}. By applying the additive property, we can deduce that this set contains
{1, 2, 3} + DTm (v, v) ⊃ {1, 2, 3} + {3k|k ≥ 0} ⊃ {1, 2, ··· , m − 1}. So we may assume that
Tm is free from directed 3-cycles, and thus a transitive tournament. Let (v0, v1, ··· , vm−1) −−→ be an ordering of Tm such that vivj if and only if i < j. In this case it is quit trivial that
(v0, vm−1) is the desired pair.
Corollary 3.1.5. For any H being an acyclic orientation of a cycle with p(H) ≥ 5, we have z(H) = p(H) + 1
Note that the case p(H) < 5 in the above corollary may result in z(H) bigger than p(H) + 1. For example when H is a transitive tournament of order 3 (p(H) = 2, z(H) = 4).
Theorem 3.1.6. Suppose H is an acyclic orientation of a union of vertex disjoint paths with the same endpoints. Then z(H) = O(p(H)d) for some constant d (independent of H).
The main tool to prove this theorem is the following lemma:
Lemma 3.1.7. Let L be the tournament consisting of two order 3 directed cycles b0, b1, b2 0 0 0 0 and b0, b1, b2 with all arcs oriented from bi to bj for 0 ≤ i, j ≤ 2. Let H be an acyclic orientation of a union of vertex disjoint paths with the same endpoints, then H → L.
Proof. Let u and v be the shared endpoints of the oriented paths in H. w.l.o.g., by choosing the right labels for the endpoints, we can assume that there is no directed path from v to u in H. We show that for any oriented path P of order k ≥ 1 with endpoints u and v (except any directed path from v to u), there is homomorphic mapping f from P to L such that 0 f(u) = b0 and f(v) = b0. Obviously by putting all such mappings together for all oriented paths P in H we get H → L. Let < u = w0, w1, ··· , wk−1, wk = v > be an arrangement of CHAPTER 3. COMPRESSIBILITY NUMBER FOR SOME ORIENTED GRAPHS 16
−−−−→ −−−−→ V (P ) where for each 1 ≤ i ≤ k either wi−1wi or wiwi−1. There should be an index r such −−−−−→ that wr−1wr, otherwise P will be a directed path from v to u. Define the oriented paths
P1,P2 to be the induced sub-oriented graphs of the subsets {w0, ··· , wr−1} and {wr, ··· , wk} in P , respectively. Clearly P1 is homomorphic to the order 3 directed cycle b0, b1, b2 in L.
Let f1 be homomorphic mapping corresponding to this homomorphism. We can assume f1(w0) = b0. Similarly let f2 be the homomorphic mapping from P2 to the order 3 directed 0 0 0 0 cycle b0, b1, b2 such that f2(wk) = b0. Then by putting f1 and f2 together we obtain the desired homomorphic mapping f from P to L.
Proof. (Theorem 3.1.6) To complete the proof we need to use some concepts and theorems expounded in Sections 4 and 5. So we postpone the proof to the end of Section 5.
Open Question 3.1.8. For which oriented graphs H, z(H) = p(H) + 1? or z(H) = O(p(H) + 1)?
The above oriented graphs are all special cases of planar graphs:
Conjecture 3.1.9. There is a constant d such that for every acyclic orientation H of a planar graph, we have: z(H) = O(p(H)d).
This conjecture will be discussed in the next chapter.
3.2 Dense graphs
A well known result of Erd˝os-Moser[8] can be translated as follows:
n−1 Theorem 3.2.1. ([8]) Suppose H is a transitive tournament with n vertices, then 2 2 ≤ z(H) ≤ 2n−1.
Proof. ([8]) By induction on n we show that every tournament of order 2n−1 contains a transitive sub-tournament of order 2n−1. For n = 1 it is trivial. Suppose the claim holds for all tournaments with order 2k−1 with k less than n ≥ 2. Let T be an arbitrary tournament on n vertices. Let v ∈ V (T ) such that |N +(v)| ≥ |N −(v)|. Such a vertex exists because P + P − + n−2 v∈V (T ) |N (v)| = v∈V (T ) |N (v)|. Then the set N (v) has a at least 2 vertices. According to the induction hypothesis it should contain a transitive sub-tournament of order n − 1. Adding the vertex v to this tournament, we arrive at a transitive sub-tournament of order n. CHAPTER 3. COMPRESSIBILITY NUMBER FOR SOME ORIENTED GRAPHS 17
Now suppose that every tournament of order m contains a transitive sub-tournament of order n. Using a probabilistic argument we will find a lower bound on m. Consider m distinct vertices. Every tournament on these vertices contains a transitive sub-tournament m of order n which must have one of the n subsets as its vertices. Any one of these subsets in order to be transitive, can be ordered in n! ways. Having fixed the transitive subset m − n (including its order) we observe that such a transitive subset can appear in exactly 2( 2 ) (2) m tournaments, since such a tournament is determined by the orientations on its 2 edges m n ( 2 ) with 2 of which have already been fixed. Finally, since each of 2 oriented graphs has a transitive sub-tournament of n vertices we have: m m − n m n!2( 2 ) (2) ≥ 2( 2 ) n
m n−1 n 2 By using n ≤ m /n! we get m ≥ 2 .
For a, b ≥ 1 the (a, b)-graph is the complete bipartite (simple) graph with a vertices on one part and b vertices on the other part. We consider acyclic orientations of the (a, b)- graph. In this case the compressibility number varies widely depending on the orientation. We attempt to find the worst case:
Definition 3.2.2. For integers a, b, define z(a, b) = max{z(H)} where the maximum is taken over all acyclic orientations H of the (a, b)-graph.
Definition 3.2.3. For every integer a, T (a, a + 1) is an orientation of the (a, a + 1)- graph with two parts A and B such that the elements of A and B can be arranged in a sequence (x1, x2, ··· , xa) and (y1, y2, ··· , ya+1) respectively with these properties: for every −−→ −−→ 1 ≤ i ≤ a + 1 and i ≤ j ≤ a, yixj and for every 1 ≤ i ≤ a and i < j ≤ a + 1, xiyj. The oriented graph T (a, a) is obtained from T (a, a + 1) by eliminating the vertex ya+1.
Theorem 3.2.4. Suppose b ≥ a ≥ 1, then z(a, b) = z(T (a, a + )) where = 1 if b > a and zero otherwise.
Proof. Let G be the (a, b)-graph with parts A (|A| = a) and B. Obviously G can be oriented in such a way that it contains T (a, a + ). So z(a, b) ≥ z(T (a, a + )). Now we show H → T (a, a + ), for any acyclic orientation H of the (a, b)-graph. H being acyclic, one can arrange its vertices in a sequence A =< v1, v2, ··· , va+b > such that there are no indexes −−→ i < j such that vjvi. In a similar way define the arrangement B =< u1, u2, ··· , u2a+ > for CHAPTER 3. COMPRESSIBILITY NUMBER FOR SOME ORIENTED GRAPHS 18 the oriented acyclic graph T (a, a + ). Now define the mapping f : V (H) → V (T (a, a + )) recursively as follows: f(v1) = u1 and for i = 2, ··· , a + b, suppose f(vi−1) = uj, then define f(vi) to be uj if vi and vi−1 both belong to the same part of H (i.e. either A or B) and uj+1 otherwise. It is easy to see that this recursion works its way to the end without encountering any problem and the resulted mapping f is indeed a homomorphic mapping from H to T (a, a + ). This implies that z(H) ≤ T (a, a + ) for every orientation H of the (a, b)-graph. Thus z(a, b) ≤ z(T (a, a + )) and by combining with the first inequality we get z(a, b) = z(T (a, a + )).
It is interesting to note that z(T (a, a + )) equals to r(T (a, a + ))) for ∈ {0, 1} (one color Ramsey number of T (a, a + ), see [12]). A similar probabilistic argument to [8] yields:
Theorem 3.2.5. z(T (a, a + )) ≥ 2a/2.
Proof. (a slight modification of the theorem 3.2.1’s proof given in [8]) Suppose every tour- nament of order m contains a T (a, a + ). We want to find a lower bound on m. Consider m distinct vertices. Every tournament on these vertices contains a T (a, a+) which must have m one of the 2a+ subsets as its vertices. Any one of these subsets in order to be a T (a, a+), first must be divided into two parts A and B, having a and b vertices respectively. This can 2a+ be done in a ways. Then each part must be ordered, this gives a!(a + )! alternatives. Having fixed the T (a, a + ) subset (including its order) we observe that such a subset can m −a(a+) appear in exactly 2( 2 ) tournaments, since such a tournament is determined by the m orientations on its 2 edges with a(a + ) of which have already been fixed. Finally, since m each of 2( 2 ) oriented graphs contains a T (a, a + ) we have: m 2a + m −a(a+) m a!(a + )!2( 2 ) ≥ 2( 2 ) 2a + a
m 2a+ a/2 By using 2a+ ≤ m /(2a + )! we get m ≥ 2 .
Conjecture 3.2.6. z(T (a, a + )) ≤ 2a+O(1) for ∈ {0, 1} Chapter 4
Relationship with Erd˝os-Hajnal conjecture
For a graph G we say the subset S ⊆ V (G) is a homogeneous set if its induced subgraph is either a clique or a graph with no edges. We denote by hom(G) the size of the largest homogeneous set in G. Similarly we define hom(T ) for a tournament T as the size of its biggest transitive sub-tournament. The classic Erd˝os-Hajnal [7] conjecture deals with the value of hom(G) when G avoids a certain forbidden graph. We say the graph G avoids another graph F , or is F -free, if it has no induced subgraph isomorphic to F (see [7]).
Definition 4.0.7. The family of monotone functions {gf : N → N}f=1,2,··· is called an Erd˝os-Hajnalfunction family if for every f ≥ 1, n ≥ 1 and every graph F with size at most f the following condition holds: For every size n F -free graph G we have hom(G) ≥ gf (n). The same definition applies to oriented Erd˝os-Hajnalfunction family, with F being an arbitrary oriented graph (with size at most f) and the notions F -free graph G and hom(G) replaced with F -free tournament T and hom(T ), respectively.
Definition 4.0.8. We say the oriented graph F is an oriented Erd˝os-Hajnal Restriction (EHR), if there exists a constant c(F ) such that for every F -free tournament T on n vertices hom(T ) ≥ nc(F ). The same definition can be applied to the case when F is a graph. In this case we say F is an EHR, if there exists a constant c(F ) such that for every F -free graph G on n vertices hom(G) ≥ nc(F ).
Based on the above definitions, the Erd˝os-Hajnal conjecture can be formulated as follows:
19 CHAPTER 4. RELATIONSHIP WITH ERDOS-HAJNAL˝ CONJECTURE 20
Conjecture 4.0.9. ([7]) Every graph is an EHR, or equivalently, for every integer f, there c(f) exists a positive integer c(f) such that {n }f=1,2,··· is an Erd˝os-Hajnalfunction family.
Alon et al., in [1], introduce the oriented version of the above conjecture:
Conjecture 4.0.10. ([1]) Every oriented graph is an oriented EHR, or equivalently, for c(f) every integer f, there exists a positive integer c(f) such that {n }f=1,2,··· is an oriented Erd˝os-Hajnalfunction family.
They proved that these two conjectures are equivalent.
Theorem 4.0.11. ([1]) Conjectures 4.0.9 and 4.0.10 are equivalent
Based on their techniques, one can come up with a slightly more generalized statement:
Theorem 4.0.12. Let {gf : N → N}f=1,2,··· be a family of monotone functions. The function 0 0 family {gf : N → N}f=1,2,··· is defined as gf = gdcf 3 log2 fe for every f ≥ 1 where c is a big 0 enough constant (during the proof we will see how big it should be). Then {gf } is an oriented 0 Erd˝os-Hajnalfunction family if {gf } is an Erd˝os-Hajnalfunction family. Similarly, {gf } is an Erd˝os-Hajnalfunction family if {gf } is an oriented Erd˝os-Hajnalfunction family.
The proof of the above theorem is based on two results from [1] and [15]:
Definition 4.0.13. A graph G with a linear order < on its vertices is called an ordered graph and denoted by (G, <). An ordered graph (G, <) is a subgraph of another ordered graph (G0, <0), if there is a mapping f : V (G) → V (G0) with these two conditions: 1. f(u) <0 f(v) if and only if u < v, 2. (f(u), f(v)) ∈ E(G0) if and only if (u, v) ∈ E(G).
Definition 4.0.14. A tournament T with a linear order < on its vertices is called an ordered tournament and denoted by (T, <). An ordered tournament (T, <) is a sub-tournament of another ordered tournament (T 0, <0), if there is a mapping f : V (T ) → V (T 0) with these −−−−−−→ two conditions: 1. f(u) <0 f(v) if and only if u < v, 2. f(u)f(v) if and only if uv−→.
Theorem 4.0.15. ([15]) For any ordered graph (G, <) on n vertices, there exists a graph G0 on O(n3 log2 n) vertices such that, for every ordering <0 of G0, (G, <) is a subgraph of (G0, <0).
Theorem 4.0.16. (The oriented version of 4.0.15 [1]) For any ordered tournament (T, <) on n vertices, there exists a tournament T 0 on O(n3 log2 n) such that, for every ordering <0 of T 0, (T, <) is a sub-tournament of (T 0, <0). CHAPTER 4. RELATIONSHIP WITH ERDOS-HAJNAL˝ CONJECTURE 21
The proofs of these two theorems are based on probabilistic method. I do not include these proofs as they are very long and the techniques are not related to this thesis. The reader may refer to the original references for the complete proofs.
Proof. (theorem 4.0.12) Suppose {gf } is an Erd˝os-Hajnal function family. Let F be an arbitrary oriented graph on f vertices. w.l.o.g. we can assume that F is a tournament.
Define an arbitrary ordering G(F, G(T, The proof of the second part is very similar. Suppose {gf } is an oriented Erd˝os-Hajnal function family and F be an arbitrary simple graph on f vertices. For an arbitrary ordering The following very general theorem shows a direct link between the ’Erd˝os-Hajnal-type’ CHAPTER 4. RELATIONSHIP WITH ERDOS-HAJNAL˝ CONJECTURE 22 problems and the compressibility number: Theorem 4.0.17. Let {gf }f=1,2,··· be an oriented Erd˝os-Hajnalfunction family and H be an acyclic oriented graph whose oriented coloring number is at most r. Define k as the smallest integer such that gr(k) ≥ p(H) + 1 Then we have z(H) ≤ k. Proof. Let T be an arbitrary tournament of size k. Since χo(H) ≤ r, there should be an oriented graph H0 on at most r vertices such that H → H0. If T contains H0 then it means H → T and we are done. If T avoids H0, then it contains a transitive sub-tournament T 0 0 with gr(k) ≥ p(H)+1 vertices. Then according to Lemma 3.1.2 H → T which again means H → T . Corollary 4.0.18. Assuming Conjecture 4.0.9 (or equivalently 4.0.10), for every class of oriented graphs H with oriented coloring number not exceeding a constant b, there is a constant d(b) such that z(H) = O(p(H)d(b)). In particular the conjecture 3.1.9 follows as planar graph orientations have bounded oriented coloring number (see [14]). c(f) Proof. Let {n }f=1,2,··· be an oriented Erd˝os-Hajnal function family as described in Con- jecture 4.0.10. Applying Theorem 4.0.17 for this function family, we get k = d(p(H) + 1)1/c(b)e and the corollary follows by placing d(b) = 1/c(b). Additionally the above corollary holds for other families of oriented graphs with bounded oriented coloring number such as oriented graphs with bounded degree, or tree width, or genus (see [11, 13, 17, 18]). Erd˝osand Hajnal, in [7], proved a weaker version of conjecture 4.0.9: Theorem 4.0.19. ([7]) For every graph F , there exists a constant c(F ) such that for every √ F -free graph G on n vertices: hom(G) ≥ 2c(F ) log n. The techniques used in the proof of this theorem are not used anywhere in this thesis. So I do not include its long full proof here and instead, I give a general outline of the proof. Please refer to the original paper [7] for the complete proof. Proof. (Theorem 4.0.19, A short summary from [7]) They define by induction a family Γ of graphs, which is called co-graphs, as follows: Initially Γ contains the graph with one vertex. Assume G1,G2 ∈ Γ. Then G ∈ Γ where V (G) = V (G1) ∪ V (G2) and E(G) is ∗ ∗ either E(G1) ∪ E(G2) or E(G1) ∪ E(G2) ∪ E , in which E = {(u1, u2): u1 ∈ V (G1), u2 ∈ CHAPTER 4. RELATIONSHIP WITH ERDOS-HAJNAL˝ CONJECTURE 23 1/2 V (G2)}. Let f = |V (F )|. A simple observation is for H ∈ Γ, hom(H) ≥ |V (H)| . Thus it is sufficient to show that every F -free graph G on n vertices contains a members √ of Γ with order at least g(n) = ec log n−K for some constant c (depending on F ) and K = (f log 2 + 2 log f)/f. This claim is proofed by induction on n. The key lemma used for this induction states that if an order n graph G avoids F , then it contains two large (in terms of n, f and a free variable used for optimization) disjoint vertex subsets A and B such that either each vertex of A is connected to many vertices of B (i.e. almost complete bipartite graph) or each vertex of A is connected to a few vertices of B (i.e. almost an empty bipartite graph). Then applying the induction hypothesis to A, we get that A contains an order g(|A|) subset A1 with its induced subgraph being in Γ. Then, based on the property of A, B, they find a large subset B1 of B which either each vertex of A1 is connected to each vertex of B1 or no vertex of A1 is connected to any vertex of B1. Now applying the induction hypothesis to B1, we get an order g(|B1|) subset B2 of B1 with its induced subgraph being in Γ. Putting A1 and B2 together gives us a member of Γ again (see the definition of Γ) of order |A1| + |B2| = g(|A|) + g(|B1|). Then the values of |A| and |B1| being large enough, they prove that g(|A|) + g(|B1|) is indeed ≥ g(n) and we are done. One can easily convert it to its oriented analogous: Theorem 4.0.20. For every oriented graph F , there exists a constant c0(F ) such that every F -free tournament T with n vertices contains a transitive sub-tournament whose size is at √ least 2c0(F ) log n Proof. Theorem 4.0.19 states that there is a function d : N → R such that, {gf (n) = √ d(f) log n d2 e}f=1,2,··· is an Erd˝os-Hajnal function family. Applying Theorem 4.0.12, we are 0 0 led to the oriented Erd˝os-Hajnal function family {gf (n)} defined as gf (n) = gdcf 3 log2 fe(n) = 2 √ d2d(dcf 3 log fe) log ne. Thus we just need to define d0(f) = d(dcf 3 log2 fe) and c0(F ) = d0(|V (F )|). √ c0(f) log n This provides us with the oriented Erd˝os-Hajnal function family {gf (n) = 2 }f=1,2,··· where c0(f) = min{c0(F )} with minimum taken over all oriented graphs F with at most f vertices. By plugging this function family into theorem 4.0.17 we get: Corollary 4.0.21. For every oriented graph H with oriented coloring number bounded by a constant b (in particular planar graphs’ orientations), there is a constant d(b) such that z(H) = O(p(H)d(b) log p(H)). CHAPTER 4. RELATIONSHIP WITH ERDOS-HAJNAL˝ CONJECTURE 24 Proof. √ log(p(H)+1) 0 log 3 g (k) = 2c (b) log k ≥ p(H) + 1 ⇒ k = d(p(H) + 1) c0(b)2 e ⇒ d(b) = b log 2 · c0(b)2 Chapter 5 Some oriented Erd˝os-Hajnal problems Theorem 4.0.17 motivates us to find as many oriented EHRs as possible. The following theorem gives a more specific relation between conjecture 3.1.9 and oriented EHRs: Theorem 5.0.22. For every family L of oriented graphs, if there exists a finite set X of oriented EHRs such that, for every H ∈ L: H → X for some X ∈ X, then there exists a d constant d such that z(H) = O(p(H) ) for every H ∈ L. Proof. For every X ∈ X let c(X) be the constant that is defined in Definition 4.0.8. Let c = minX∈X{c(X)} and define d = d1/ce. For an arbitrary H ∈ L, let X ∈ X be the oriented graph such that H → X. Let T be any tournament of order (p(H) + 1)d. If T contains X, then H → T . Otherwise, X being an oriented EHR, T contains a transitive sub-tournament T 0 of order |V (T )|c(X) ≥ p(H) + 1. Then according to Lemma 3.1.2, H will be homeomorph to T 0 and thus H → T again. Open Question 5.0.23. Is there any finite set X of oriented EHRs such that, for every acyclic orientation H of a planar graph, H → X with X ∈ X(The affirmative answer to this question immediately implies Conjecture 3.1.9). 5.1 Partitioning restrictions Definition 5.1.1. The vertex cut (A, B) of the oriented graph H, with none of A and B being empty, is called an oriented cut, if all the arcs of the cut, if there is any, are oriented 25 CHAPTER 5. SOME ORIENTED ERDOS-HAJNAL˝ PROBLEMS 26 from A to B. Definition 5.1.2. The oriented graph F of order f ≥ 3 is called a partitioning restriction if every F -free tournament of order at least f has an oriented cut. For example every directed cycle is a partitioning restriction (see theorem 5.1.4). The implication is: Theorem 5.1.3. Every partitioning restriction is an oriented EHR. Proof. By induction on n we show that every tournament T with n ≥ 2 vertices avoiding a partitioning restriction F of order f ≥ 2, contains a transitive sub-tournament of order n b f c + 2 and thus F should be an oriented EHR. It’s easy to check it when n < f, since every such tournament has a 2-transitive sub-tournament. Now suppose T is of order at least f and the claim holds for all smaller tournaments. Since T avoids F , according to the definition of F , T has an oriented cut (A, B). Let T1 and T2 be the largest transitive sub-tournament of the sub-tournaments induced by A and B respectively. Let T ∗ be the ∗ induced sub-tournament of V (T1) ∪ V (T2). Trivially T is transitive. w.l.o.g. assume V (T1) ≥ V (T2) ≥ 1. If |V (T2)| = 1 then we must have |B| = 1. This means |A| = n − 1 ≥ 2. Thus by applying the induction hypothesis to the induced graph of A we get: ∗ n−1 n |V (T )| = |V (T1)| + |V (T2)| ≥ b f c + 2 + 1 ≥ b f c + 2. Now suppose |V (T2)| ≥ 2. The apply the induction hypothesis to both induced graphs of A and B, we’ll have: |V (T ∗)| = |A| |B| n |V (T1)| + |V (T2)| ≥ b f c + 2 + b f c + 2 ≥ b f c + 2. So the inductive argument follows. Here we introduce one family of partitioning restrictions: Theorem 5.1.4. Every directed cycle C of order c ≥ 3 is a partitioning restriction. Proof. Let T be a tournament of order n ≥ 2 avoiding C. If c = 3 then C is transitive and trivially we are done. So we may assume that c ≥ 4 and T has a cycle of length 3. Let k be the largest number ≤ c, such that T contains a cycle K of order k. Obviously k < c. Let v0, v1, ··· , vk−1 be the vertices of K in the clockwise order. For each v ∈ V (T ) − V (K), either N +(v) ∩ V (K) = ∅ or N −(v) ∩ V (K) = ∅. For otherwise, there are x, y ∈ V (K) such that xv−→ and vy−→. K being a cycle, we can easily deduce that there is an index 0 ≤ i < k −→ −−−→ such that viv and vvi+1 (index calculations are all mod k). Then we have the directed cycle + − v, vi+1, vi+2, ··· vi of order k+1. Contradicting the definition of k. Let A and A be the set of vertices v for which N −(v)∩V (K) = ∅ and N +(v)∩V (K) = ∅, respectively. The previous CHAPTER 5. SOME ORIENTED ERDOS-HAJNAL˝ PROBLEMS 27 argument states A+∪A− = V (T )−V (K). We claim that all the arcs between A+ and A− are oriented from the former to the latter. Otherwise there are u ∈ A+ and w ∈ A− such that −→ wu. Then the directed cycle u, vk−1, vk−2, ··· , v1, w has length k +1. Again the definition of k is violated. Either A+ or A− is nonempty, since |A+ ∪ A−| = |V (T ) − V (K)| = n − k ≥ 1. Call the nonempty one A. Then it’s easy to see that either (A, V (T ) − A) or (V (T ) − A, A) is an oriented cut. Corollary 5.1.5. Every directed cycle is an oriented EHR. Corollary 5.1.6. Every oriented graph F with at most four vertices is an oriented EHR. Proof. W.L.G we can assume F is a tournament. It is trivial when |V (F )| ≤ 2. When |V (F )| = 3, F is either a transitive tournament (an obvious oriented EHR) or a directed cycle. In the latter case c(F ) = 1. Now suppose |V (F )| = 4. Suppose F contains a vertex v with either in-degree or out-degree zero. Let F 0 be obtained from F by deleting v. Let A be the single arc oriented graph. Then F is isomorphic to A(v, F 0) (refer to 5.2.1 for the definition). Knowing that both A and F 0 are oriented EHRs, by applying Theorem 5.2.3 (in the next subsection) we conclude that so is F . Let T be an arbitrary tournament on n vertices. If it avoids the order 4 directed cycle, then according to Corollary 5.1.5 it contains a transitive sub-tournament of order nc for some constant c. Now suppose T contains a −−−−→ directed cycle with vertex set X = {x0, x1, x2, x3} where xixi+1 for all i = 0, ··· , 3 (index calculations are mod 4). Then if F does not contain such a vertex v, a simple argument shows that F → T |X . Thus we can set c(F ) = c. 5.2 A constructive method A powerful constructive method in [1] enables us to make new EHRs, out of the known EHRs. Definition 5.2.1. For any graph G with V (G) = {v1, v2, ··· , vk} and k other graphs F1,F2, ··· ,Fk we define G(F1,F2, ··· ,Fk) as the graph obtained from G by replacing each vi with a copy of Fi, and joining any vertex of the copy of Fi to any vertex of a copy of Fj , j 6= i, if and only if vivj ∈ E(H). Similarly we define H(F1,F2, ··· ,Fk), for oriented graphs H,F1,F2, ··· ,Fk with V (H) = {v1, v2, ··· , vk}, by replacing each vi with a copy of Fi and joining each vertex in Fi to each vertex Fj, i 6= j, with an arc oriented from the former set −−→ to the latter set if and only if vivj. CHAPTER 5. SOME ORIENTED ERDOS-HAJNAL˝ PROBLEMS 28 Theorem 5.2.2. ([1]) If G, F1, ··· ,Fk are simple EHRs, then so is G(F1,F2, ··· ,Fk). Similar to theorem 4.0.20, one can come up with its oriented version: Theorem 5.2.3. If H,F1, ··· ,Fk are oriented EHRs, then so is H(F1,F2, ··· ,Fk). The proof of the above theorem is easily obtained from the proof of Theorem 5.2.2 by replacing the terminologies for graphs with that of oriented graphs. More precisely, for a vertex v in a graph, the notions of set {u : u and v are adjacent} and set {u : u and v are not adjacent} must be replaced with the notions N +(v) and N −(v), respec- tively in the oriented version of the proof (in which v will be a vertex in an oriented graph). Proof. (Theorem 5.2.3. Oriented version of the theorem 5.2.2’s proof given in [1]) Let V (H) = {v1, v2, ··· , vk}. Obviously, it is sufficient to show that H(F1, v2, ··· , vk) is an oriented EHR. Let H0 denote the oriented graph obtained from H by the deletion of v1. For simplicity, write H(F1) for H(F1, v2, ··· , vk). Let T be an H(F1)-free tournament with n vertices, and assume that hom(T ) < nc(H)δ (c(H) is the constant defined in Definition 4.0.8). We would like to get a contradiction, provided that δ > 0 is sufficiently small. Let m = dnδe > k. By the definition of c(H), any m-element subset of U ⊆ V (T ) must contain H. Otherwise, we would find a transitive sub-tournament of order > mc(H) which n n−k is impossible. Therefore, T has at least m / m−k induced sub-oriented graphs isomorphic to H. For each of these sub-graphs, fix an isomorphic mapping from H into T . Since the number of mappings from H0 to T is smaller than n(n − 1) ··· (n − k + 2), there exists a mapping, which can be extended to an isomorphic mapping of H to T in at least n M = m (5.1) n−k m−k n(n − 1) ··· (n − k + 2) 0 0 different ways. In other words, there are k − 1 vertices v2, ··· , vk ∈ V (T ), and there exists an at least M-element subset W ⊆ V (T ) such that, for every w ∈ W : 0 f(v1) = w, f(vi) = vi(i = 2, ··· , k) is an isomorphic mapping from H to T . Consider now the sub-tournament T |W of T induced by W . This tournament must be F1-free, otherwise T would not be H(F1)-free. Since F1 an oriented EHR, we know that the order of c(F1) c(F1) hom(T |W ) ≥ |W | ≥ M CHAPTER 5. SOME ORIENTED ERDOS-HAJNAL˝ PROBLEMS 29 On the other hand: c(H)δ n > hom(T ) ≥ hom(T |W ) Comparing the last two inequalities and plugging in the value (5.1) for M, we obtain that: n n − k + 1 nδc(H)/c(F ) > m = > n1−kδ n−k m(m − 1) ··· (m − k + 1) m−k n(n − 1) ··· (n − k + 2) which gives the desired contradiction, provided that: c(F ) δ < c(H) + k · c(F ) Corollary 5.2.4. Every tournament F , with the order of its largest directed cycle at most four, is an oriented EHR. Proof. Induction on the order of F . For |V (F )| ≤ 4 the claim holds according to Corollary 5.1.6. Assume |V (F )| ≥ 5. Since F avoids the length 5 directed cycle, then according to Theorem 5.1.4 F contains an oriented cut (B,C). B and C are both tournaments with order smaller than the order of F and contain no directed cycle larger than 4. So according to the induction hypothesis both are oriented EHRs. Let A be the single arc oriented graph. Then applying Theorem 5.2.3 we conclude that F = A(B,C) (Definition 5.2.1) is also an oriented EHR. Conjecture 5.2.5. Every oriented graph F , with the order of its largest directed cycle less than four, is an oriented EHR. Now we can give the proof of Theorem 3.1.6: Proof. (Theorem 3.1.6) Let A and B be the single arc oriented graph and the order 3 directed cycle, respectively. Then we observe that the tournament L defined in Lemma 3.1.7 is exactly A(B,B) (Definition 5.2.1). Note that both A and B are oriented EHRs, so according to Theorem 5.2.3, L = A(B,B) is an oriented EHR too. Now applying Theorem 5.0.22, for the family of oriented graphs H and X = {L}, guarantees the existence of such constant d. Bibliography [1] Noga Alon, J´anosPach, and J´ozsefSolymosi. Ramsey-type theorems with forbidden subgraphs. Combinatorica, 21(2):155–170, 2001. [2] W.G. Brown, P. Erd¨os,and M. Simonovits. Extremal problems for directed graphs. J. Combinatorial Theory Ser. B, 15:77–93, 1973. [3] W.G. Brown, P. Erd¨os,and M. Simonovits. Inverse extremal digraph problems. Colloq. Math. Soc. Jnos Bolyai, 37, I,II:119–156, 1984. [4] W.G. Brown, P. Erd¨os,and M. Simonovits. Algorithmic solution of extremal digraph problems. Trans. Amer. Math. Soc., 292(2):421–449, 1985. [5] W.G. Brown and F. Harary. Extremal digraphs. Coll. Math. Soc. J. Bolyai, 4, 1969. [6] B. Courcelle. The monadic second order-logic of graphs vi : On several representations of graphs by relational structures. Discrete Appl. Math., 54:117–149, 1994. [7] P. Erd¨osand A. Hajnal. Ramsey-type theorems. Discrete Appl. Math, 25:37–52, 1989. [8] P. Erd¨osand L. Moser. On the representation of directed graphs as unions of orderings. Math. Inst. Hung. Acad. Sci, 9:125–132, 1964. [9] P. Erd¨osand M. Simonovits. A limit theorem in graph theory. Studia Sci. Math. Hungar, 1:51–57, 1966. [10] P. Erd¨osand A.H. Stone. On the structure of linear graphs. Bull. Amer. Math. Soc., 52:1087–1091, 1946. [11] A.V. Kostochka, E.´ Sopena, and X. Zhu. Acyclic and oriented chromatic numbers of graphs. J. Graph Theory, 24:331–340, 1997. [12] Yannis Manoussakis and Zsolt Tuza. Ramsey numbers for tournaments. Theor. Com- put. Sci., 263(1-2):75–85, 2001. [13] P. Ochem and A. Pinlou. Oriented colorings of partial 2-trees. Inform. Process. Lett., 2008. 30 BIBLIOGRAPHY 31 [14] A. Raspaud and E.´ Sopena. Good and semi-strong colorings of oriented planar graphs. Inform. Process. Lett., 51(4):171–174, 1994. [15] Vojtech R¨odland Peter Winkler. A ramsey-type theorem for orderings of a graph. SIAM J. Discrete Math., 2(3):402–406, 1989. [16] Alexander Sidorenko. Boundedness of optimal matrices in extremal multigraph and digraph problems. Combinatorica, 13(1):109–120, 1993. [17] E.´ Sopena. The chromatic number of oriented graphs. J. Graph Theory, 25:191–205, 1997. [18] E.´ Sopena. Oriented graph coloring. Discrete Math., 229(1-3):359–369, 2001. [19] P. Tur´an. On an extremal problem in graph theory. Mat. Fiz . Lapok, 48:436–452, 1941. [20] P. Tur´an.On the theory of graphs. Colloq . Math, 3:19–30, 1954.